Proof. Consider two distinct graphs \(G_1\)  and \(G_2\) , which are not isospectral but have equal connectivity. Let, the Laplacian matrices corresponding to the graphs \(G_1\) , \(G_2\)  are \(L_1\) , \(L_2\) , respectively. Here, we study a specific case where different Laplacians not only share the same 2nd smallest eigenvalue or algebraic connectivity but also share the corresponding eigenvector or Fiedler vector [1]}. The eigenvector equations of two Laplacian matrices are given by
\(& (L_1 - \lambda _2 I ). \mathbf {v}_F = \mathbf {0}\\ & (L_2 - \lambda _2 I ). \mathbf {v}_F = \mathbf {0}\) 
